In the times of the Bais Hamikdash, were judges appointed by the king, or were they elected by the people? And was the system different based on which size court (i.e., courts comprised of 3, 23, or 71 judges)?

In the mishnah in one of the first few chapters of Sanhedrin (or maybe a commentary there), it says that there were 71 students at the sanhedrin, and when one member of the sanhedrin got knocked out the greatest student took his place
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b aAug 31 '12 at 16:14

@ba It's Sanhedrin 4:4. But that doesn't explain how those students got there to begin with.
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Double AA♦Aug 31 '12 at 18:04

@DoubleAA I'm guessing that when G-d told Mosheh to gather 70 zekeinim, they also gathered 71 students (but there's no source for that, as far as I know)
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b aAug 31 '12 at 18:06

2

@ba That's very possible, but I think the question was more about how that population is maintained over time. Meaning, when those students became judges, who became the new students?
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Double AA♦Aug 31 '12 at 18:08

to add to Double AA's Comment: and did the same system apply to the local beis din or was that just for the sanheidrin?
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shachnaAug 31 '12 at 18:46

2 Answers
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From Rambam, Hilchos Sanhedrin Ch. 2 (Chabad.org) it appears that, for the most part, the appointment of judges was the responsibility of the Supreme Sanhedrin:

Halacha 8
Our Sages relate: From the Supreme Sanhedrin, they would send emissaries throughout the entire land of Israel to seek out judges. Whenever they found a person who was wise, sin-fearing, humble, modest, with a good reputation, and beloved by people at large, they have him appointed as a judge in his own city. From there, they promote him to the court which holds sessions at the entrance to the Temple Mount. From there, he is promoted to the court which holds sessions at the entrance to the Temple Courtyard, and from there, to the Supreme Sanhedrin.

It seems that judges could also be appointed by kings, and afterward, exilarchs. Ch.3

Halacha 8
Whenever a Sanhedrin, a king, or an exilarch appoints a judge who is not fitting and/or is not learned in the wisdom of the Torah and is not suitable to be a judge....

Chapter 5 (Halacha 1) implies that the king's power of appointment was limited to courts of 3:

A minor Sanhedrin [of 23 judges] for every tribe and every city may be appointed only by the High Court of 71 judges.

Judges received their authority from their immediate predecessors who "laid their hands" upon them, a process known as "semicha." The president of the Great Sanhedrin was the authority who conferred judicial powers on graduating judges in a formal procedure before a court of three. Judges were, however, also appointed by kings, a power which appears to have eventually devolved with the rule of Babylonia.